Abstract
In any calculation in perturbative Quantum Chromodynamics (QCD) a choice needs to be made for the unphysical renormalisation scale, $\mu_R$. The Brodsky-Lepage-Mackenzie/Principle of Maximum Conformality (BLM/PMC) scale-setting procedure is one proposed method for selecting this scale. In this work we identify three ambiguities in the BLM/PMC procedure itself. Their numerical impact is studied using the example of the total cross-section for $t\bar{t}$ production through Next-to-Next-to-Leading Order in QCD. One ambiguity is the arbitrary choice of the value of the highest-order PMC scale. The numerical impact of this choice on the BLM/PMC prediction for the cross-section is found to be comparable to the impact of the choice of $\mu_R$ in the conventional scale-setting approach. Another ambiguity relates to the definitions of the other PMC scales and is similarly found to have a large impact on the cross-section.
Highlights
When performing calculations in quantum chromodynamics (QCD), any partonic observable ρ is usually calculated as a perturbative series in the strong coupling constant αS:ρ 1⁄4 Xn cnðμRÞαS4ðμπRÞn: ð1ÞThe renormalization scale μR is an arbitrary parameter which enters this equation following renormalization.Formally, when working to all orders in αS, the μR dependence of the coefficients cn exactly compensates that of αS so that ρ is independent of μR
We find that the subtraction of the Coulomb terms has a large impact on the PMC scales q2 and q3; failure to subtract the Coulomb terms leads to q3 ≪ ΛQCD and a divergent cross section
In order to do so, we recompute the above cross sections using a variety of choices for the PMC scales (q2, q3, q4): (1) (q2, q3, q3) First, we restate the results using the “standard” choice of PMC scales which were used in the previous section. (2) (q2, q3, mt) Second, we study the numerical impact of the arbitrary choice of the scale q4 by setting q4 1⁄4 mt rather than q4 1⁄4 q3. (3) (q02; q03; q03) Third, we explore the other main ambiguity discussed in Sec
Summary
When performing calculations in quantum chromodynamics (QCD), any partonic observable ρ is usually calculated as a perturbative series in the strong coupling constant αS:. Not essential for the goals of the present work, we would like to mention that more refined arguments for choosing this scale have been given in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] Such arguments tend to modify the choice μR 1⁄4 Q by a factor of Oð1Þ and are especially relevant for observables with several kinematic scales. We expect that many of our findings transcend this particular process
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